Biharmonic hypersurfaces with constant scalar curvature in space forms

Abstract

Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form ${\mathbb{M}}^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\le 0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space ${\mathbb{E}}^{n+1}$ or hyperbolic space ${\mathbb{H}}^{n+1}$ for $n<7$.

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